cost optimized design technique for pseudo-random numbers in cellular automata

7/31/2019 Cost Optimized Design Technique for Pseudo-Random Numbers in Cellular Automata

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International Journal of Advanced Information Technology (IJAIT) Vol. 2, No.3, June 2012

DOI : 10.5121/ijait.2012.2302 21

Cost Optimized Design Technique for

Pseudo-Random Numbers in Cellular Automata

Arnab Mitra1, 3 and Anirban Kundu2, 3

1 Adamas Institute of Technology, West Bengal - 700126, [email protected]

2 Kuang-Chi Institute of Advanced Technology, Shenzhen - 518057, [email protected]

3 Innovation Research Lab (IRL), West Bengal - 711103, India

[email protected]

ABSTRACT

In this research work, we have put an emphasis on the cost effective design approach for high quality

pseudo-random numbers using one dimensional Cellular Automata (CA) over Maximum Length CA. This

work focuses on different complexities e.g., space complexity, time complexity, design complexity and

searching complexity for the generation of pseudo-random numbers in CA. The optimization procedure for

these associated complexities is commonly referred as the cost effective generation approach for pseudo-

random numbers. The mathematical approach for proposed methodology over the existing maximum length

CA emphasizes on better flexibility to fault coverage. The randomness quality of the generated patterns for

the proposed methodology has been verified using Diehard Tests which reflects that the randomness qualityachieved for proposed methodology is equal to the quality of randomness of the patterns generated by the

maximum length cellular automata. The cost effectiveness results a cheap hardware implementation for the

concerned pseudo-random pattern generator. Short version of this paper has been published in [1].

KEYWORDS

Pseudo-Random Number Generator (PRNG), Prohibited Pattern Set (PPS) & Fault Coverage,

Randomness Quality, Diehard Tests, Cellular Automata (CA)

1. INTRODUCTION

Production of random patterns, plays a fundamental role in the field of research work varyingfrom Computer Science, Mathematics or Statistics to cutting edge VLSI Circuit testing. Random

numbers are also used in cryptographic key generation and game playing. Mathematicians

describe that this random numbers happen in a sequence where the values are homogeneously

distributed over a well defined interval, and it is unfeasible to predict the next values based on its

past or present ones.
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]


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In probabilistic approach, a random number [2] is defined as a number, chosen as if by chancefrom some precise distribution such that selection of a large set of these numbers reproduces the

fundamental distribution. Those numbers are also required to be independent to maintain no

correlations between successive numbers. In the generation process of random numbers over

some specified boundary, it is often essential to normalize the distributions such that eachdifferential area is equally populated. In order to generate a power-law distribution P(x) from a

uniform distribution P(y), it is P(x)= Cxn

for x [x0,x1]. Then normalization follows as Equation

(1),

=+=

+1

0

1

0

1

11

][)(

x

x

x

x

n

n

xCdxxP .. (1)

From Equation (1), the value of C can be determined as followed in Equation 2.

We get, 10

1

1

1++

+=

nn

xx

nC .. (2)

In practice, random number generator requires a specification of an initial number used as the

starting point, which is known as a "seed". Random number generators are classified into several

groups based on the difference in generation procedures of random numbers. Pseudo-random

number and true-random number are most commonly used in scientific works. The classification

of the random number is based upon the selection of seed, that describes how much randomized

way has been adopted for the selection of that seed for generation of random pattern. The qualityof any random number generator is being tested through a statistical test suit named Diehard Tests

[3].

Random numbers or patterns [2], [4] obtained by means of executing a computer program, is

based on implementing a particular recursive algorithm are commonly referred to as pseudo-random numbers. Pseudo, emphasizes that the selection of seed is in deterministic way.

The quality of randomness generated by any random number generator is needed to be verified.

For this purpose, the diehard tests are a battery of statistical tests for measuring the quality of a

random number generator. This statistical test suit was developed by George Marsaglia over

several years and first published in 1995 on a CD-ROM of random numbers [3]. The following

tests are executed as enlisted below to measure the quality of the randomness of any particularrandom pattern generator:

i) Birthday Spacings: This name is based on the birthday paradox. Here a pair of random pointson a large interval is chosen such that the spacing between the points should be asymptotically

exponentially distributed.

ii) Overlapping Permutations: In this test, the sequences of five consecutive random numbers

are analyzed to observe overlapping of the permutations; a total 120 possible orderings should

occur with statistically equal probability.

iii) Ranks of Matrices: In this test, some number of bits are selected from some number of


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random numbers to form a matrix over {0, 1}. Thereafter the rank is counted based on thedeterminant value of the matrices.

iv) Monkey tests: This name is based on the infinite monkey theorem. Here the sequences of

some number of bits are assumed as "words". Thereafter the count is completed over the

overlapping words in a stream. The number of "words" that does not appear should follow anyknown distribution.

v) Count the 1s: In this test, the 1 bit in each of either successive or chosen bytes is counted and

then the counted value is converted into "letters" and finally following the counting of the

occurrences of five-letter "words".

vi) Parking Lot Test: In this test, randomly placed unit circles in a 100 x 100 square are tested to

find whether any of the circles, overlaps an existing one. After an iteration of 12000 tries, thenumber of successfully "parked" circles should follow a certain normal distribution.

vii) Minimum Distance Test: In this test, the minimum distance between the pairs of randomly

placed 8,000 points in a 10,000 x 10,000 square is measured. The square of this distance is

exponentially distributed with a certain mean.

viii) Random Spheres Test: In this test, randomly chosen 4000 points with the center a sphere on

each point is placed in a cube of edge 1000 and the radius is calculated. The smallest sphere's

volume should be exponentially distributed with a certain mean.

ix) The Squeeze Test: In this test a multiplication is performed with 231 by random floats on [0,

1) until 1 is achieved. The iteration of this multiplication occurs around 100000 times. Thenumber of floats needed to reach 1 should follow a certain distribution.

x) Overlapping Sums Test: In this test a long sequence of random floats on [0, 1) is generated.Thereafter an addition of sequences of 100 consecutive floats is performed. The sums should be

normally distributed with characteristic mean and sigma.

xi) Runs Test: In this test a long sequence of random floats on [0, 1) is generated to perform the

ascending and descending runs tests. The counts should follow a certain distribution.

xii) The Craps Test: In this test the games of craps is played for 200000 times to obtain the wins

and the number of throws per game. Each count should follow a certain distribution.

Cellular Automata [5, 6] is used to represent a dynamic mathematical model. A CellularAutomaton (pl. cellular automata, in short CA) is a discrete model studied in computability

theory, mathematics, physics, complexity science, theoretical biology and microstructure

modeling. It consists of a regular grid of cells, each in one of a finite number of states, such as "1"

for ON and "0" for OFF. The framework might be in higher degree of dimensions. For each cell, aset of cells called its neighborhood (usually including the cell itself) is defined relative to the

specified cell. For example, the neighborhood of a cell might be defined as the set of cells a

distance of 2 or less from the cell. A primary state (time t=0) is selected by assigning a state for

each cell. A new generation is created (advancing t by 1), according to some predetermined

mathematical function that determines the new state of each cell in terms of the current state ofthe cell and the states of the cells in its neighborhood.


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The simplest CA is one-dimensional with only two neighbors defined at the adjacent cells oneither side of it. This combination forms a neighborhood of 3 cells, with 23=8 possible patterns

for this neighborhood. There are then 28=256 possible rules. These 256 CAs are generally

referred to by their Wolfram code. Wolfram code is a standard naming convention invented by

Wolfram; it gives each rule a number from 0 to 255. A number of papers have analyzed andcompared these 256 CAs, either individually or collectively.

CA has a significant role for computation as it is easy to implement through hardware

components. Following Figure 1 shows a rough outline of CA implementation and Figure 2

describes the hardware implementation of CA through flip-flop.

Figure 1. Implementation of 3-cell Null Boundary CA

Figure 2. Hardware implementation of CA

Rest of the paper is organized as follows: Section 2 briefs about the related work; Section 3

describes proposed work; Experimental observations and result analysis are shown in Section 4

and Conclusion is reflected in Section 5.

2. RELATED WORK

Several methods have been implemented to generate good quality random numbers. Some novel

efforts have been established to generate good quality random numbers [5], [7-19]. Some

important efforts have also been made to generate pseudo-random numbers using CA. It has beenalready established that the pseudo-random numbers generated using maximum length CA

exhibits a high degree of randomness.

The most common way to generate pseudo-random number is to use a combination of

randomize and rand functions. Random patterns can be achieved based on the following

recursive PRNG Equation 3.


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)(211 ModNPXPX nn +=+ .........(3)

Here P1, P2 are prime numbers, N is range for random numbers. Xn is calculated recursively using

the base value X0. X0 is termed as seed and it is a prime number.

If X0 (seed) is same all time or selected in any deterministic way, then it yields pseudo-randomnumber [2].

It has been observed that in the generation process of pseudo-random pattern using MaximumLength CA, only a large cycle is responsible for yielding the pseudo random patterns. If a single

cycle with larger numbers of states is used to generate the random patterns, the associated cost

would be increased. All the cost associated with the generation process of random numbers; i.e.,time, design and searching costs are having higher values as the complexity is directly

proportional to the number of cell sizes used in the cycle. So, it is convenient to reduce the cycle

without affecting the randomness quality of the generated random patterns for reducing these

associated costs.

For the generation process of high quality of pseudo-random numbers, several works have beenestablished with the uses of Cellular Automata. The research on Maximum Length CA reveals

that with the increased number of cells, the resulting maximum length cycle includes the

maximum degree of randomness. In Maximum Length CA, prohibited pattern set (PPS) isexcluded from the cycle for achieving higher degree of randomness. In Maximum Length CA, the

generation procedure of random patterns is complex and costly. In this methodology, it is

mandatory to keep track of PPS in maximum length cycle and to exclude the portion in resultingmaximum length cycle. Thus it makes the generation procedure complex. The associated

complexity costs i.e., time complexity, design complexity and searching complexity are high

along with this procedure. The quality of randomness in generated integer pattern is quite high

[1]. Therefore, an alternative easy to implement generation methodology of random numbers

would be much more beneficial which deal with the flaws of Maximum Length CA. In ourproposed methodology, we have proposed a system that will be dealing with the flaws of the

Maximum Length CA. Thus the proposed methodology should be more cost effective in terms of

design complexity, time complexity and searching complexity though the generated random

integer patterns are having the quality of randomness, is at least equal to the quality ofrandomness achieved by Maximum Length CA. The detailed discussion of the proposed

methodology is following in Section 3.

3. PROPOSED WORK

A cost effective and simpler design methodology always should be beneficial for the generation

of random integer patterns. The cost efficiency refers to the space, time, searching and design

complexity of any involved algorithm. In the cost optimization generation methodology of theuniversal random pattern, a one dimensional Equal Length Cellular Automata, is proposed overthe existing Maximum Length Cellular Automata for random pattern generation.

In the proposed methodology, a new approach has been proposed to achieve high degree of

randomization in terms of better cost optimization with respect to all the concerning complexities

mentioned earlier. The resulting methodology should also be very much convenient in hardware


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implementation. The quality of randomness of the generated pattern by the proposedmethodology is compared through Diehard Tests with Maximum Length CA random number

generator. The conclusion is made about the quality of randomness of any data set based on the

number of Diehard Tests passed. Several data set are analyzed through Diehard, is described in

Table 4 to compare for randomness quality of different random number generators.

In the proposed PRNG, we propose an equal length one dimensional CA over the maximum

length CA. In our work, we propose the decomposition of the larger cycle of maximum length

CA, into more relevant sub-cycles such that our concerning complexities cost can be reduced as

well as the fault coverage can be more flexible than of previously established maximum lengthCA. The proposed PRNG system is described in Figure 3 [1].

Figure 3. Flowchart of proposed system

A new mathematical approach has been proposed to achieve the same amount of randomization in

less cost with respect to various complexities and hardware implementation, according to the

flowchart of the proposed system as mentioned in Figure 3.

The resulting larger cycle of maximum length CA is divided into two or more equal sub-cycles

instead of taking the full cycle of maximum length. These sub-cycles are capable to generate

random patterns as equivalent to the randomness quality achieved from maximum length CA. The

following Algorithm 1 has been used for the decomposition of an n-cell maximum length CA [1].


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Algorithm 1: Cycle Decomposition

Input: CA size (n), PPS Set

Output: m-length cycles excluding PPS

Step 1: Start

Step 2: Initialize the number of n-cell CA to generate random patterns using n-cell CA

Step 3: Decompose the cell number (n) into two equal numbers (m) such that n=2*m

Step 4: Check each PPS whether it belongs to a single smaller cycle CA

Step 5: Repeat Step 3 and Step 4 until each PPS belongs to separate smaller cycles

Step 6: Allow m-length cycles of n-cell CA after excluding all the PPS containing cycles

Step 7: Stop

In the proposed methodology, the primary concern is to reduce the PPS. Therefore it has been

taken care of that the occurrence of every PPS must be completed in some of the smaller sub-

cycles, such that by eliminating all those smaller cycles, the remaining prohibited patterns free

cycles can be allowed to generate random patterns. Thus, this methodology implies a better cost

effectiveness approach. The proposed methodology thus simplifies the design complexity and

empowers the searching complexity. The terminology design complexity refers to the

implementation procedure for generation of random pattern and empowering searching

complexity means the zero overhead for keeping track for PPS for random pattern generation. In

comparison with an n-cell maximum length CA, more number of smaller cycles instead of onemaximum length cycle should be used.

The maximum length CA cycle has been decomposed into smaller equal length cycles usingAlgorithm 1. The Algorithm 1 explains the decomposition procedure of maximum length CA.

Let us assume, for the n-length CA, the total number of states= 2n.

We can generate this same amount of CA cells using Equation 4.

We have 2n= 2*(2m); i.e., at least 4 numbers of equal length cycles.

Thus m is always less than n.

Let one maximum length CA is divided into two sub-cycles.

Then, 2n = (2m1 + 2m2)

If m1= m2

Then it becomes 2n = 2 (2m1 )

Upon illustrating this model, let us assume the case where maximum length CA, n=8.

So, 28=27+27 (i.e., two equal length of CA cycles)

=26+26 +26 +26 (i.e., four equal length of CA cycles)

=25+2

5+2

5+2

5+2

5+2

5+2

5+2

5(i.e., eight equal length of CA cycles)

=24+24+24+24+24+24+24+24+24+24+24+24+24+24+24+24 (i.e., sixteen equal length of CA

cycles)

The PPS is excluded from the cycle as per the procedure for generating the maximum random

pattern in maximum length CA. In our procedure, the PPS can be totally removed as we are

having more number of cycles for generation of random sequences. In proposed methodology, the

cycles producing PPS can be removed from the generation of pseudo random numbers.


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According to the proposed methodology, a CA size of n=24 might be decomposed into equallength smaller cycles instead of one maximum length circle. In this case it can be divided into 16

smaller cycles of length 20 or more. Let us assume that there exist nine number prohibited

patterns as PPS. So in worst case, assuming every single prohibited pattern occurs in a single

cycle, there exist (16 - 10=6) number of 20 CA cycle to generate random patterns. The patterngeneration on this scenario is followed as in Figure 4. Figure 4(a) shows one maximum length

cycle with prohibited patters and on the hand Figure 4(b) shows 16 equal length smaller cycles

where some the cycles contains prohibited set only. The following Figure 4 is based on Null

Boundary 3 cell CA where the rules have been applied. The PPS is denoted as

{PS0, PS1PS9}.

Figure 4(a).

Figure 4(b).

Figure 4. Figure. 4(a). Maximum length CA Cycle for n=24 and Figure 4(b). Proposed equal length CA of

smaller cycle size


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Further a CA size of n=64 might be decomposed into equal length smaller cycles instead of onemaximum length cycle. In this case it can be decomposed it into 128 smaller cycles of length 57

or more. Let us assume that there exist hundred number prohibited patterns as PPS. So in worst

case, every single prohibited pattern will be in a single cycle, there exist (128 -100=28) number

of cycles of length 57 to generate random patterns. The pattern generation is followed as inFigure 5. Figure 5(a) shows one maximum length cycle with prohibited patters and Figure 5(b)

shows 100 equal length smaller cycles contains prohibited pattern only. Figure 5 is based on Null

Boundary 3 cell CA where the rules have been applied. The PPS is denoted as

{PS0, PS1PS99}. Thus higher numbers of PPS are discarded from random pattern generating

equal length CA cycles.

Figure 5(a).

Figure 5(b).

Figure 5. Figure. 5(a). Maximum length CA Cycle for n=64 and Figure 5(b). Proposed equal length CA of

smaller cycle size


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In terms of fault coverage, the maximum length CA follows a procedure to handle the PPS if itexists in the larger cycle, which is responsible for generating random pattern. The fault coverage

mechanism is as followed in following section.

In practice, the maximum length CA based random number generator produces the randompatterns of integer using the cycle which might contain some of the prohibited patterns. In this

scenario, the PPS are excluded form that maximum length CA cycle. Assume that there exist

some n-numbers of PPS in the maximum length cycle. Let the PPS are {PS 0,,PSn }. For the

exclusion purpose of these PPS from the cycle, the minimum length of arc (Arc min) between the

prohibited pattern PS1 and PSn should be measured so that we can utilize the remaining cycle arci.e. effective arc (Earc) for random number generation, which is typically free from PPS. This

scenario can be efficiently visualized in Figure 6.

Figure 6. Typical Cycle structure of an n-cell Maximum Length CA

The Figure 6 is based on the facts that there exists total n-number of prohibited patterns with the

following set: PPS={PS1,..,PSn}. Figure 6 also explain the procedure to measure Arcmin and Earcfrom an n-cell maximum length CA for fault coverage in random pattern generation.

Definition 1: The minimum length of arc (Arcmin) is the minimum distance between the first andlast prohibited pattern in an n-cell maximum length CA cycle.

Definition 2: The effective arc (Earc) is the remaining arc length of an n-cell maximum length CAcycle which excludes Arcminfrom the corresponding CA cycle of states and it is responsible for

generating pseudo random patterns of integers.

In our proposed methodology, we have discarded all the cycles containing any prohibited pattern.Some equal length cycles containing PPS are being discarded, yet we have enough cycles for

generation of random patterns. The following Table-1 shows the comparison of fault coveragebetween maximum length CA and our proposed CA for CA cell size n=9. Let us assume that thereexists 9 numbers of prohibited patterns. This procedure is described in Figure 7.


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Figure 7. Typical cycle structure for maximum length CA cycle for n=9

4. EXPERIMENTAL OBSERVATIONS AND RESULT ANALYSIS

The PPS containing arc in random pattern generating cycle is excluded from the cycle as per the

procedure for generating the maximum random pattern in maximum length CA. On the other

hand, the PPS containing cycles are totally removed to generate the random sequences. There isno overhead to calculate Earc and Arcmin in proposed equal length CA methodology. The following

Table 1 compares the procedures of maximum length CA and equal length CA based random

pattern generators.

Table 1: Comparison of fault coverage procedures

Maximum length CA Equal Length CA

PPS 9 9

Earc 503 N/AArcmin 9 N/A

Table 1 reflects the advantages of our proposed methodology over maximum length CA for fault

coverage in random pattern generation for CA size n=9. In proposed equal length CA, the cycles

containing any prohibited pattern is excluded from generating random patterns.

The p-value analysis, generated for each sample data set by Diehard battery series test, helps to

decide whether the test data set passes or fails the diehard test. Diehard returns the p-value, which

should be uniform [0, 1) if the input file contains truly independent random bits. Those p-values

are obtained by p=F(x), where F is the assumed distribution of the sample random variable x,which is often normal. The value p0.975 means the RNG has failed the test at the

0.05 level. Table 2 indicates the different tests of Diehard Test Suit.


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Table 2: Diehard Tests

No. Name of the test

1 Birthday Spacings

2 Overlapping Permutations3 Ranks of 31x31 and 32x32 matrices

4 Ranks of 6x8 Matrices

5 The Bitstream Test

6 Monkey Tests OPSO,OQSO,DNA

7 Count the 1`s in a Stream of Bytes

8 Count the 1`s in Speci_c Bytes

9 Parking Lot Test

10 Minimum Distance Test

11 The 3DSpheres Test

12 The Sqeeze Test

13 Overlapping Sums Test

14 Runs Test15 The Craps Test

The comparison result in Table 3 shows the degree of randomness achieved by different random

number generators in Diehard Tests.

Table 3: Performance result through Diehard for different n-values in maximum length CA-RNG

Diehard Test

Number

Max-length CAn=8

Max-length CAn=10

Max-length CAn=23

Max-length CAn=64

1 Fail Fail Pass Pass




5 Fail Fail Fail Fail

6 Fail Fail Fail Pass








14 Fail Fail Pass Pass15 Fail Fail Pass Pass

Total Number of

Diehard Test

Passes

0 0 10 14


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Diehard Tests

0

5

10

15

n=8 n=10 n=23 n=64

No. of Cells in Max Length CA

No.ofDiehard

TestsPass

ed

Randomness Quality

Figure 8. Randomness Quality for Maximum Length CA for different cell sizes (n)

Table 3 shows the degree of randomness achieved in Maximum Length CA for different cell sizes

in terms of the total number of Diehard Tests passes. Figure 8 reflects this degree of randomness

graphically.

Table 4: Performance result through Diehard for different CA random number generators

Diehard Test Number Max-length CA Equal length CA

n=23 n=64 n=23 n=64

1 Pass Pass Pass Pass




5 Fail Fail Fail Fail

6 Fail Pass Fail Pass










Total Number of Diehard

Test Passes

10 14 10 14


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Figure 9. Diehard Performance Graph

The results obtained from Table 4 and Figure 9 ensures that the proposed random number

generator, posses the maximum degree of randomization with a reference to the number ofDiehard Test passes. This result is similar to the result achieved for maximum length CA based

random number generator. The various complexities of these two CA based methodologies are

reflected in Table 4.

Table 5: Complexity Comparison between Maximum Length CA and Proposed Methodology

Name of the

Complexity

Comparison Result

Space Same

Time Slightly Improved

Design Improved

Searching Improved

Table 5 signifies that the space complexity is same for both procedures as total length of an n-cell

CA is same for both the cases, but there are some changes in other complexities. Other

complexities have been improved in case of our proposed methodology. The proposedmethodology is allowed only to generate random patterns from smaller cycles that exclude PPS.

The PPS exclusion feature from the main cycle, improves the design and searching complexities.

Results based on Table 4, Table 5 and Figure 8, it can be concluded that our proposed

methodology can be used as a better source of random patterns as it posses the maximum degree

of randomness with reference to all other established RNGs and it is less costly to implement.


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5. CONCLUSION

The Diehard Test results show the quality of randomness achieved from the various samples of

random data sets. The number of passes shows the quality of randomness achieved by particular

method. Table 4 shows the randomness achieved from the random data sets produced throughEqual Length CA is having the maximum randomness as it passes the maximum number of

Diehard Tests which is same as for the maximum length CA. Hence this methodology is suitable

for generating random sequences as the cost associated is much cheaper in terms of time

complexity and hardware implementation. The Equal Length CA uses only the sub cycle whichconsists of lesser number of states than of maximum length CA. Thus it completes its full cycle in

lesser time and requires lesser hardware implementation cost. The time complexity of this

proposed methodology maintains a linear time complexity with increased number of cells in the

CA. It is also convenient that the Equal Length CA random number generator is more flexible as

it excludes the prohibited pattern set (PPS).

REFERENCES

[1] Arnab Mitra, Anirban Kundu; (2012) Cost optimized Approach to Random Numbers in Cellular

Automata; The Second International Conference on Computer Science, Engineering & Applications

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duke.edu/~rgb/General/dieharder.php. C program archive dieharder, version 1.4.24.

[4] Dirk Eddelbuettel; Random: An R package for true random

numbers;http://dirk.eddelbuettel.com/bio/papers.html

[5] http://www.random.org/

[6] S. Wolfram; (1986) Theory and Application of Cellular Automata; World Scientific.

[7] Sukanta Das, Biplab K. Sikdar, P. Pal Chaudhuri; (2004) Characterization of Reachable/Nonreachable Cellular Automata States; International Conference on Cellular Automata for Research

and Industry; ACRI; Netherlands.

[8] Sukanta Das, Anirban Kundu, Biplab K. Sikdar, P. Pal Chaudhuri; (2005) Design of Nonlinear CA

Based TPG Without Prohibited Pattern Set In Linear Time; JOURNAL OF ELECTRICAL TESTING:

Theory and Applications.

[9] Biplab K. Sikdar, Sukanta Das, Samir Roy, Niloy Ganguly, Debesh K. Das; (2005) Cellular Automata

Based Test Structures with Logic Folding; VLSI Design; India.

[10] Sukanta Das, Haffizur Rahaman and Biplab K Sikdar; (2005) Cost Optimal Design of NonlinearCA

Based PRPG for Test Applications; IEEE 14th Asian Test Symposium; India.

[10] Sukanta Das, Debdas Dey, Subhayan Sen, Biplab K. Sikdar, Parimal Pal Chaudhuri; (2004) An

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[12] Sukanta Das, Anirban Kundu, Biplab K. Sikdar; (2004) Nonlinear CA Based Design of Test Set

Generator Targeting Pseudo-Random Pattern Resistant Faults; Asian Test Symposium; Taiwan.

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Authors:

Mr. Arnab Mitra

Mr. Mitra is presently working as an Assistant Professor in the Department of CSE with

Adamas Institute of Technology-W.B. (India). He is pursuing his research work as a

Research Assistant at Innovation Research Lab, under guidance of Dr. Anirban Kundu.

His broad research interests are Artificial Intelligence and applications of CellularAutomata.

Dr. Anirban Kundu

Dr. Kundu is a Post Doctoral Research Fellow at Kuang-Chi Institute of Advanced

Technology, P.R. China. He has obtained his Ph.D. from Jadavpur University (India).

In India he is associated with Netaji Subhash Engineering College as an Assistant

Professor & Head of the Department (IT). He is also associated with Innovation

Research Lab-W.B. (India) as an Advisor & Consultant. His research interests include

Search Engine Oriented Indexing, Ranking, Prediction, Web Page Classification with

essence of Cellular Automata. He is also interested in Multi-Agent based system

design, Fuzzy controlled systems and Cloud Computing. He has authored more than 40 journal and

conference papers.

cost optimized design technique for pseudo-random numbers in cellular automata

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